Transcript how to find midpoint - Bremen High School District 228

CHAPTER 2
TEST
REVIEW
Segment Bisectors:
The midpoint of a segment is the point on the segment that divides it into two
congruent segments.
A segment bisector is a segment, ray, line, or plane that intersects a segment at
Its midpoint.
To bisect a segment means to divide the segment into two congruent segments.
Examples:
●
●
●
A
M
B
M is midpoint of AB.
Examples:
1.
Find AM and MB
38
●
●
●
A
M
B
2.
Find MH and GH
●
●
●
G
M
H
18
3.
Find x.
●
J
5x- 9
●
M
16
●
K
HOW TO FIND MIDPOINT:
1. (7,-8) and (9,2)
2. (-14,7) and (-4,-15)
3. (-6,-10) and (-4,-3)
ANGLE BISECTORS:
An angle bisector is a ray that divides an angle into two angles that are congruent.
●
A
●
D
BD bisects
ABD
●
B
●
C
ABC
DBC
Examples:
HK bisects
1.
GHJ. Find the m
G
●
GHK and m
KHJ.
2.
●K
●
J
●
K
64°
●
J
145°
●
H
●
H
●
G
3.
H
●
●
J
●
K
G●
4.
●K
●
J
●
H
●
G
Find x.
H●
7.
2x + 11
J
●
8.
G●
53°
G●
K
●
K●
6x
H●
4x + 8
What is the m
What is the m
●
J
GHK and m
GHJ.
KHJ.
COMPLEMENTARY AND SUPPLEMENTARY ANGLES:
Two angles are complementary angles if the sum of their measure is 90°
Two angles are supplementary angles if the sum of their measures is 180°
1
2
Angles 1 and 2 are supplementary.
3
4
Angles 3 and 4 are complementary.
Determine whether the angles are complementary, supplementary or neither.
1.
2.
68°
132°
22°
48°
41°
3.
48°
4.
145°
42°
Measures of compliments and supplements:
1.
A and
B are complements. If m
A = 23° find m
B.
2.
C and
D are supplements. If m
C = 113° find m
D.
3.
E and
F are supplements. If m
E = 39° find m
F.
VERTICAL ANGLES:
Two angles are vertical angles if they are not adjacent and their sides are
formed by two intersecting lines.
2
1
4
3
1 and
3 are vertical angles
2 and
4 are vertical angles
Examples:
1. Find m
1
2. Find m
2
3. Find m
3
2
1
3
68°
4. Find x.
5. Find m
1
6. Find m
2
2x + 67
1
2
4x + 63
Two adjacent angles are a linear pair if their noncommon sides are on the same line.
common
side
5
noncommon
side
5 and
6
noncommon
side
6 are a linear pair
EXAMPLES:
1. Find x.
x
81°
y
136°
2. Find y.
3. Find x.
4. Find m
D●
ABD
●
A
2x + 33
●
B
53°
●
C
IF-THEN STATEMENTS AND DEDUCTIVE REASONING:
An if-then statement has two parts. The “if” part contains the
hypothesis. The “then” part contains the conclusion.
If a number is divisible by 2 then the number is even.
HYPOTHESIS
CONCLUSION
EXAMPLES:
Identify the hypothesis and the conclusion.
1. If it rains today then the game will be cancelled.
2. If angle is 120° then it is obtuse.
Write if-then statements:
1. I will buy the cell phone if it costs less then $50.
2. You need to take the ACT test your junior year of high school.
Example:
Use the law of syllogism to write an if-then statement
for the following pair of statements.
If the perimeter of a square is 24 ft, then
the length of a side of the square is 6 ft.
If the length of a side of a square is 6 ft, then
the area of the square is 36 square feet.
PROPERTIES OF EQUALITY AND CONGRUENCE:
PROPERTIES OF EQUALITY AND CONGRUENCE
Reflexive Property
Equality AB = AB
m A=m
Congruence
A
Symmetric Property
Equality
If AB = CD then CD = AB
If m A = m B then m B = m A
AB ≅ AB
A≅ A
Congruence
If AB ≅ CD then CD ≅ AB
If A ≅ B then B ≅ A
Transitive property
Equality
If AB = CD and CD = EF,
then AB = EF.
Congruence
If AB ≅ CD and CD ≅ EF,
then AB ≅ EF.
If m A = m B and m B = m C,
then m A = m C.
If A ≅ B and
then A ≅ C
B≅
C,
Use properties of equality:
Addition Property:
Adding the same number to each side of an equation
produces an equivalent equation.
x–3=7
x -3 +3=7+3
Subtraction Property:
Subtracting the same number from each side of an
equation produces an equivalent equation.
y + 5 = 11
y + 5 – 5 = 11 – 5
Multiplication Property:
Multiplying each side of an equation by the same
nonzero number produces an equivalent equation.
x=6
x●4=6●4
Division Property:
Dividing each side of an equation by the same
nonzero number produces an equivalent equation.
8x = 16
8𝑥 16
=
8
8
Substitution Property:
Substituting a number for a variable in an equation
produces an equivalent equation.
x=7
2x + 4 = 2(7) + 4
Homework
Pages 95-97
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